Group #14524921 — subgroups

Subgroups — C₁₇×#1915 #14524921

Order 2176.

308 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₇×#568	C₂	1	1	yes	maximal, characteristic
2	C₁₇×#536	C₂	1	1	yes	maximal, characteristic
3	C₁₇×#413	C₂	1	1	yes	maximal, characteristic
4	C₂×C₁₇×#101	C₂	1	2	yes	maximal, complemented
5	C₁₇×#344	C₂	1	2	yes	maximal, complemented
6	C₂×#41×C₁₇	C₂²	1	1	yes	characteristic
7	C₂²×Q₈×C₁₇	C₂²	1	1	yes	characteristic
8	C₁₇×#131	—	2	4	no	—
9	C₂×M₁₆×C₁₇	—	2	2	no	—
10	C₂×M₁₆×C₁₇	C₂²	1	2	yes	—
11	C₂×M₁₆×C₁₇	C₂²	1	2	yes	—
12	C₁₇×C₄⋊Q₈	—	2	1	no	—
13	C₁₇×#124	—	2	1	no	—
14	C₁₇×#123	C₄	1	4	yes	—
15	C₁₇×C₂²⋊Q₈	—	2	2	no	—
16	C₁₇×#117	C₂²	1	1	yes	characteristic
17	C₁₇×#101	—	2	1	no	—
18	C₁₇×#101	—	2	4	no	—
19	C₁₇×#100	—	2	1	no	—
20	#41×C₁₇	—	2	2	no	—
21	#41×C₁₇	—	2	2	no	—
22	C₂×Q₈×C₁₇	—	2	1	no	—
23	C₂×Q₈×C₁₇	—	2	1	no	—
24	C₂×Q₈×C₁₇	—	2	2	no	—
25	C₂×Q₈×C₁₇	—	2	2	no	—
26	C₂×Q₈×C₁₇	D₄	1	1	yes	characteristic
27	C₂×D₄×C₁₇	C₂×C₄	1	2	yes	—
28	C₂×D₄×C₁₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₁₇	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₁₇	D₄	1	2	yes	—
31	C₂²×C₄×C₁₇	C₂³	1	1	yes	characteristic
32	M₁₆×C₁₇	—	2	4	no	—
33	M₁₆×C₁₇	—	2	2	no	—
34	M₁₆×C₁₇	—	2	2	no	—
35	M₁₆×C₁₇	—	2	2	no	—
36	M₁₆×C₁₇	D₄	1	4	yes	—
37	C₂×C₈×C₁₇	—	2	4	no	—
38	C₂×C₈×C₁₇	—	2	2	no	—
39	C₂×C₈×C₁₇	—	2	2	no	—
40	C₄⋊C₄×C₁₇	—	2	4	no	—
41	C₄⋊C₄×C₁₇	—	2	1	no	—
42	C₂²⋊C₄×C₁₇	—	2	4	no	—
43	C₂²⋊C₄×C₁₇	C₂×C₄	1	2	yes	—
44	C₄²×C₁₇	—	2	1	no	—
45	C₂³×C₁₇	C₂²⋊C₄	1	2	yes	—
46	C₂³×C₁₇	C₂²×C₄	1	1	yes	characteristic
47	Q₈×C₁₇	—	2	2	no	—
48	Q₈×C₁₇	—	4	1	no	—
49	Q₈×C₁₇	—	4	2	no	—
50	Q₈×C₁₇	—	4	1	no	—
51	D₄×C₁₇	—	2	4	no	—
52	D₄×C₁₇	—	2	2	no	—
53	C₂×C₄×C₁₇	—	2	2	no	—
54	C₂×C₄×C₁₇	—	2	2	no	—
55	C₂×C₄×C₁₇	—	2	2	no	—
56	C₂×C₄×C₁₇	—	2	2	no	—
57	C₂×C₄×C₁₇	—	2	4	no	—
58	C₂×C₄×C₁₇	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₁₇	C₂×D₄	1	2	yes	—
60	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
61	C₂×C₄×C₁₇	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₇	#41	1	2	yes	—
63	C₈×C₁₇	—	2	4	no	—
64	C₈×C₁₇	—	4	2	no	—
65	#1915	C₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂²×C₁₇	—	2	2	no	—
67	C₂²×C₁₇	—	4	1	no	—
68	C₂²×C₁₇	—	4	2	no	—
69	C₂²×C₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₁₇	C₄×D₄	1	2	yes	—
71	C₄×C₁₇	—	2	2	no	—
72	C₄×C₁₇	—	4	2	no	—
73	C₄×C₁₇	—	4	2	no	—
74	C₄×C₁₇	#120	1	1	yes	characteristic
75	C₄×C₁₇	#121	1	2	yes	—
76	C₄×C₁₇	#123	1	1	yes	characteristic
77	#568	C₂×C₁₇	1	1	yes	characteristic
78	#536	C₂×C₁₇	1	1	yes	characteristic
79	#413	C₂×C₁₇	1	1	yes	characteristic
80	C₂×#101	C₂×C₁₇	1	2	yes	complemented
81	#344	C₂×C₁₇	1	2	yes	complemented
82	C₂×C₁₇	—	2	1	no	—
83	C₂×C₁₇	—	2	2	no	—
84	C₂×C₁₇	—	4	2	no	—
85	C₂×C₁₇	#386	1	1	yes	characteristic
86	C₂×#41	C₂²×C₁₇	1	1	yes	characteristic
87	C₂²×Q₈	C₂²×C₁₇	1	1	yes	characteristic
88	#131	—	2	4	no	—
89	C₂×M₁₆	—	2	2	no	—
90	C₂×M₁₆	C₂²×C₁₇	1	2	yes	—
91	C₂×M₁₆	C₂²×C₁₇	1	2	yes	—
92	C₄⋊Q₈	—	2	1	no	—
93	#124	—	2	1	no	—
94	#123	C₄×C₁₇	1	4	yes	—
95	C₂²⋊Q₈	—	2	2	no	—
96	#117	C₂²×C₁₇	1	1	yes	characteristic
97	#101	—	2	1	no	—
98	#101	—	2	4	no	—
99	#100	—	2	1	no	—
100	C₁₇	#1915	1	1	yes	characteristic, direct product factor, Sylow
101	#41	—	2	2	no	—
102	#41	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×Q₈	—	2	2	no	—
107	C₂×Q₈	D₄×C₁₇	1	1	yes	characteristic
108	C₂×D₄	C₂×C₄×C₁₇	1	2	yes	—
109	C₂×D₄	D₄×C₁₇	1	1	yes	characteristic
110	C₂²×C₄	C₂×C₄×C₁₇	1	2	yes	—
111	C₂²×C₄	D₄×C₁₇	1	2	yes	—
112	C₂²×C₄	C₂³×C₁₇	1	1	yes	characteristic
113	M₁₆	—	2	4	no	—
114	M₁₆	—	2	2	no	—
115	M₁₆	—	2	2	no	—
116	M₁₆	—	2	2	no	—
117	M₁₆	D₄×C₁₇	1	4	yes	—
118	C₂×C₈	—	2	4	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	2	no	—
121	C₄⋊C₄	—	2	4	no	—
122	C₄⋊C₄	—	2	1	no	—
123	C₂²⋊C₄	—	2	4	no	—
124	C₂²⋊C₄	C₂×C₄×C₁₇	1	2	yes	—
125	C₄²	—	2	1	no	—
126	C₂³	C₂²⋊C₄×C₁₇	1	2	yes	—
127	C₂³	C₂²×C₄×C₁₇	1	1	yes	characteristic
128	Q₈	—	2	2	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	4	1	no	—
131	Q₈	—	4	2	no	—
132	D₄	—	2	4	no	—
133	D₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	C₂²⋊C₄×C₁₇	1	2	yes	—
140	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
141	C₂×C₄	C₂×D₄×C₁₇	1	2	yes	—
142	C₂×C₄	C₂×D₄×C₁₇	1	1	yes	characteristic
143	C₂×C₄	#41×C₁₇	1	2	yes	—
144	C₈	—	2	4	no	—
145	C₈	—	4	2	no	—
146	C₂²	—	2	2	no	—
147	C₂²	—	4	1	no	—
148	C₂²	—	4	2	no	—
149	C₂²	C₂×C₂²⋊C₄×C₁₇	1	1	yes	characteristic
150	C₂²	C₄×D₄×C₁₇	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	2	no	—
154	C₄	C₁₇×#120	1	1	yes	characteristic
155	C₄	C₁₇×#121	1	2	yes	—
156	C₄	C₁₇×#123	1	1	yes	characteristic
157	C₂	—	2	1	no	—
158	C₂	—	2	2	no	—
159	C₂	—	4	2	no	—
160	C₂	C₁₇×#386	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₁₇×#568

C₂

yes

maximal, characteristic

C₁₇×#536

C₂

yes

maximal, characteristic

C₁₇×#413

C₂

yes

maximal, characteristic

C₂×C₁₇×#101

C₂

yes

maximal, complemented

C₁₇×#344

C₂

yes

maximal, complemented

C₂×#41×C₁₇

C₂²

yes

characteristic

C₂²×Q₈×C₁₇

C₂²

yes

characteristic

C₁₇×#131

—

C₂×M₁₆×C₁₇

—

C₂×M₁₆×C₁₇

C₂²

yes

—

C₂×M₁₆×C₁₇

C₂²

yes

—

C₁₇×C₄⋊Q₈

—

C₁₇×#124

—

C₁₇×#123

C₄

yes

—

C₁₇×C₂²⋊Q₈

—

C₁₇×#117

C₂²

yes

characteristic

C₁₇×#101

—

C₁₇×#101

—

C₁₇×#100

—

#41×C₁₇

—

#41×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

—

C₂×Q₈×C₁₇

D₄

yes

characteristic

C₂×D₄×C₁₇

C₂×C₄

yes

—

C₂×D₄×C₁₇

D₄

yes

characteristic

C₂²×C₄×C₁₇

C₂×C₄

yes

—

C₂²×C₄×C₁₇

D₄

yes

—

C₂²×C₄×C₁₇

C₂³

yes

characteristic

M₁₆×C₁₇

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

M₁₆×C₁₇

—

M₁₆×C₁₇

D₄

yes

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₂×C₈×C₁₇

—

C₄⋊C₄×C₁₇

—

C₄⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

—

C₂²⋊C₄×C₁₇

C₂×C₄

yes

—

C₄²×C₁₇

—

C₂³×C₁₇

C₂²⋊C₄

yes

—

C₂³×C₁₇

C₂²×C₄

yes

characteristic

Q₈×C₁₇

—

Q₈×C₁₇

—

Q₈×C₁₇

—

Q₈×C₁₇

—

D₄×C₁₇

—

D₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

—

C₂×C₄×C₁₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

—

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₇

#41

yes

—

C₈×C₁₇

—

C₈×C₁₇

—

#1915

C₁₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

—

C₂²×C₁₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₇

C₄×D₄

yes

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

—

C₄×C₁₇

#120

yes

characteristic

C₄×C₁₇

#121

yes

—

C₄×C₁₇

#123

yes

characteristic

#568

C₂×C₁₇

yes

characteristic

#536

C₂×C₁₇

yes

characteristic

#413

C₂×C₁₇

yes

characteristic

C₂×#101

C₂×C₁₇

yes

complemented

#344

C₂×C₁₇

yes

complemented

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

—

C₂×C₁₇

#386

yes

characteristic

C₂×#41

C₂²×C₁₇

yes

characteristic

C₂²×Q₈

C₂²×C₁₇

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂²×C₁₇

yes

—

C₂×M₁₆

C₂²×C₁₇

yes

—

C₄⋊Q₈

—

#124

—

#123

C₄×C₁₇

yes

—

C₂²⋊Q₈

—

#117

C₂²×C₁₇

yes

—

—

—

100

C₁₇

#1915

yes

characteristic, direct product factor, Sylow

101

#41

—

102

#41

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

D₄×C₁₇

yes

characteristic

108

C₂×D₄

C₂×C₄×C₁₇

yes

—

109

C₂×D₄

D₄×C₁₇

yes

characteristic

110

C₂²×C₄

C₂×C₄×C₁₇

yes

—

111

C₂²×C₄

D₄×C₁₇

yes

—

112

C₂²×C₄

C₂³×C₁₇

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

—

117

M₁₆

D₄×C₁₇

yes

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

C₂×C₄×C₁₇

yes

—

125

C₄²

—

126

C₂³

C₂²⋊C₄×C₁₇

yes

—

127

C₂³

C₂²×C₄×C₁₇

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂²⋊C₄×C₁₇

yes

—

140

C₂×C₄

C₂×D₄×C₁₇

yes

characteristic

141

C₂×C₄

C₂×D₄×C₁₇

yes

—

142

C₂×C₄

C₂×D₄×C₁₇

yes

characteristic

143

C₂×C₄

#41×C₁₇

yes

—

144

C₈

—

145

C₈

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂×C₂²⋊C₄×C₁₇

yes

characteristic

150

C₂²

C₄×D₄×C₁₇

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

C₁₇×#120

yes

characteristic

155

C₄

C₁₇×#121

yes

—

156

C₄

C₁₇×#123

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

C₁₇×#386

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₁₇	#386
commutator	C₂³	C₂²×C₄×C₁₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₇
prime radical	C₂×#41×C₁₇	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₇ ◁ C₂²×C₄×C₁₇ ◁ G

subgroup

G/H

center, socle

C₂×C₁₇

#386

commutator

C₂³

C₂²×C₄×C₁₇

Frattini, cosocle

C₂²×C₄

C₂³×C₁₇

prime radical

C₂×#41×C₁₇

C₂²

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₁₇ ◁ C₂²×C₄×C₁₇ ◁ G

N	H ≅ G/N	type
4. C₂×C₁₇×#101	C₂	semidirect
5. C₁₇×#344	C₂	semidirect
65. #1915	C₁₇	direct (N × H)
80. C₂×#101	C₂×C₁₇	semidirect
81. #344	C₂×C₁₇	semidirect
100. C₁₇	#1915	direct (N × H)

H ≅ G/N

type

4. C₂×C₁₇×#101

C₂

semidirect

5. C₁₇×#344

C₂

semidirect

65. #1915

C₁₇

direct (N × H)

80. C₂×#101

C₂×C₁₇

semidirect

81. #344

C₂×C₁₇

semidirect

100. C₁₇

#1915

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₇×#568	2
C₁₇×#536	2
C₁₇×#413	2
C₂×C₁₇×#101	2
C₁₇×#344	2
#1915	17

subgroup

index

C₁₇×#568

C₁₇×#536

C₁₇×#413

C₂×C₁₇×#101

C₁₇×#344

#1915

Subgroups — C₁₇×#1915 #14524921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C17×#1915 #14524921

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₇×#1915 #14524921